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Properties

Label 2-140-140.107-c1-0-15
Degree $2$
Conductor $140$
Sign $0.980 + 0.197i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 + 0.441i)2-s + (0.322 − 1.20i)3-s + (1.60 + 1.18i)4-s + (−0.381 − 2.20i)5-s + (0.965 − 1.47i)6-s + (−2.60 + 0.451i)7-s + (1.63 + 2.30i)8-s + (1.25 + 0.722i)9-s + (0.460 − 3.12i)10-s + (−0.824 + 0.476i)11-s + (1.94 − 1.55i)12-s + (−3.15 + 3.15i)13-s + (−3.70 − 0.544i)14-s + (−2.77 − 0.251i)15-s + (1.18 + 3.82i)16-s + (−0.688 + 2.56i)17-s + ⋯
L(s)  = 1  + (0.949 + 0.312i)2-s + (0.186 − 0.695i)3-s + (0.804 + 0.593i)4-s + (−0.170 − 0.985i)5-s + (0.394 − 0.602i)6-s + (−0.985 + 0.170i)7-s + (0.579 + 0.815i)8-s + (0.417 + 0.240i)9-s + (0.145 − 0.989i)10-s + (−0.248 + 0.143i)11-s + (0.562 − 0.448i)12-s + (−0.875 + 0.875i)13-s + (−0.989 − 0.145i)14-s + (−0.716 − 0.0649i)15-s + (0.295 + 0.955i)16-s + (−0.166 + 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.980 + 0.197i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.980 + 0.197i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75644 - 0.175408i\)
\(L(\frac12)\) \(\approx\) \(1.75644 - 0.175408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.441i)T \)
5 \( 1 + (0.381 + 2.20i)T \)
7 \( 1 + (2.60 - 0.451i)T \)
good3 \( 1 + (-0.322 + 1.20i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.824 - 0.476i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.15 - 3.15i)T - 13iT^{2} \)
17 \( 1 + (0.688 - 2.56i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.99 + 3.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.528 + 0.141i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.23iT - 29T^{2} \)
31 \( 1 + (3.41 - 1.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.00 - 0.269i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.71T + 41T^{2} \)
43 \( 1 + (8.73 + 8.73i)T + 43iT^{2} \)
47 \( 1 + (0.830 + 3.09i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.83 - 0.760i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-6.30 - 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.01 + 5.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-14.3 - 3.85i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.60iT - 71T^{2} \)
73 \( 1 + (14.2 + 3.80i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.34 + 9.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.72 - 5.72i)T + 83iT^{2} \)
89 \( 1 + (-1.83 - 1.05i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.08 + 3.08i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.16170977112642389609486499798, −12.42783837189369482193475994180, −11.69721659198438490832990810045, −10.01611623250552250515052366866, −8.719537818199346848455917456236, −7.48737044493679115624520029703, −6.67029813357527180092812396153, −5.27187182240469887957794615931, −4.07588905381443419236020456506, −2.23800330495145838765323398138, 2.91626448318781514947800572574, 3.70184794267636575900838365429, 5.21034773405664226867344412120, 6.58114564672018401275963413039, 7.48562605495768766708396027543, 9.660454057687272418978541045981, 10.16416579728517266714434730113, 11.10115146441021269806761617980, 12.34904796039783677357946357702, 13.12892122208869804511179309336

Graph of the $Z$-function along the critical line